The higher the model # in the family the more powerful it is. Example HD 5570 < HD 5670 < HD 5750.
Generally speaking newer generation cards are more powerful than older generation cards. Example HD 4xxx < HD 5xxx.
Comparing performance of different generation cards takes research. Here is a brief simple snapshot:
HD 4650 = HD 5550 < HD 4670 = HD 5570 < HD 5670 < HD 4770 < HD 4850 = HD 5750 < HD 4870 = HD 5770 < HD 4890 = HD 5830 (roughly speaking) < HD 5850 < HD 5870
The faster the video card the more it is able to take advantage of 1GB of RAM. The higher the resolution of the monitor, the more memory you should have. Things to keep in mind:
1. Starting at 1680 x 1050 1GB of Video RAM can make a difference depending on the game.
2. The HD 5670 is too slow to take advantage of 1GB of RAM unless you can see roughly 1% - 2% of increased performance. Likewise, if the HD 5670 can't take advantage of 1GB of RAM, slower cards will not benefit either.
ATI performance is largely determined by the number of stream processing units.
After that probably memory bandwidth which is a combination of memory type, speed and bus width.
Basically just look at the chart he linked rather than trying figure these things out for yourself based on raw stats. That chart is based on performance and doesn't factor in other things like power efficiency and dx11 compatibility. Only the HD5000 cards are DX11 and they are also usually much more power efficient than the HD4000 series.